Basic Math Examples

$\frac{s^{2} - 1}{2} + \frac{s + 2}{3} = 8$

Step 1

Simplify $\frac{s^{2} - 1}{2} + \frac{s + 2}{3}$ .

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{s^{2} - 1^{2}}{2} + \frac{s + 2}{3} = 8$

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = s$ and $b = 1$ .

$\frac{(s + 1) (s - 1)}{2} + \frac{s + 2}{3} = 8$

Step 1.2

To write $\frac{(s + 1) (s - 1)}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{(s + 1) (s - 1)}{2} \cdot \frac{3}{3} + \frac{s + 2}{3} = 8$

Step 1.3

To write $\frac{s + 2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{(s + 1) (s - 1)}{2} \cdot \frac{3}{3} + \frac{s + 2}{3} \cdot \frac{2}{2} = 8$

Step 1.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{(s + 1) (s - 1)}{2}$ by $\frac{3}{3}$ .

$\frac{(s + 1) (s - 1) \cdot 3}{2 \cdot 3} + \frac{s + 2}{3} \cdot \frac{2}{2} = 8$

Step 1.4.2

Multiply $2$ by $3$ .

$\frac{(s + 1) (s - 1) \cdot 3}{6} + \frac{s + 2}{3} \cdot \frac{2}{2} = 8$

Step 1.4.3

Multiply $\frac{s + 2}{3}$ by $\frac{2}{2}$ .

$\frac{(s + 1) (s - 1) \cdot 3}{6} + \frac{(s + 2) \cdot 2}{3 \cdot 2} = 8$

Step 1.4.4

Multiply $3$ by $2$ .

$\frac{(s + 1) (s - 1) \cdot 3}{6} + \frac{(s + 2) \cdot 2}{6} = 8$

Step 1.5

Combine the numerators over the common denominator.

$\frac{(s + 1) (s - 1) \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Expand $(s + 1) (s - 1)$ using the FOIL Method.

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Step 1.6.1.1

Apply the distributive property.

$\frac{(s (s - 1) + 1 (s - 1)) \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.1.2

Apply the distributive property.

$\frac{(s \cdot s + s \cdot - 1 + 1 (s - 1)) \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.1.3

Apply the distributive property.

$\frac{(s \cdot s + s \cdot - 1 + 1 s + 1 \cdot - 1) \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.2

Simplify and combine like terms.

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Step 1.6.2.1

Simplify each term.

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Step 1.6.2.1.1

Multiply $s$ by $s$ .

$\frac{(s^{2} + s \cdot - 1 + 1 s + 1 \cdot - 1) \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.2.1.2

Move $- 1$ to the left of $s$ .

$\frac{(s^{2} - 1 \cdot s + 1 s + 1 \cdot - 1) \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.2.1.3

Rewrite $- 1 s$ as $- s$ .

$\frac{(s^{2} - s + 1 s + 1 \cdot - 1) \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.2.1.4

Multiply $s$ by $1$ .

$\frac{(s^{2} - s + s + 1 \cdot - 1) \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.2.1.5

Multiply $- 1$ by $1$ .

$\frac{(s^{2} - s + s - 1) \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.2.2

Add $- s$ and $s$ .

$\frac{(s^{2} + 0 - 1) \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.2.3

Add $s^{2}$ and $0$ .

$\frac{(s^{2} - 1) \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.3

Apply the distributive property.

$\frac{s^{2} \cdot 3 - 1 \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.4

Move $3$ to the left of $s^{2}$ .

$\frac{3 \cdot s^{2} - 1 \cdot 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.5

Multiply $- 1$ by $3$ .

$\frac{3 \cdot s^{2} - 3 + (s + 2) \cdot 2}{6} = 8$

Step 1.6.6

Apply the distributive property.

$\frac{3 s^{2} - 3 + s \cdot 2 + 2 \cdot 2}{6} = 8$

Step 1.6.7

Move $2$ to the left of $s$ .

$\frac{3 s^{2} - 3 + 2 \cdot s + 2 \cdot 2}{6} = 8$

Step 1.6.8

Multiply $2$ by $2$ .

$\frac{3 s^{2} - 3 + 2 \cdot s + 4}{6} = 8$

Step 1.6.9

Add $- 3$ and $4$ .

$\frac{3 s^{2} + 2 s + 1}{6} = 8$

Step 2

Multiply both sides by $6$ .

$\frac{3 s^{2} + 2 s + 1}{6} \cdot 6 = 8 \cdot 6$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $6$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{3 s^{2} + 2 s + 1}{6} \cdot 6 = 8 \cdot 6$

Step 3.1.1.2

Rewrite the expression.

$3 s^{2} + 2 s + 1 = 8 \cdot 6$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $8$ by $6$ .

$3 s^{2} + 2 s + 1 = 48$

Step 4

Solve for $s$ .

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Step 4.1

Move all terms to the left side of the equation and simplify.

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Step 4.1.1

Subtract $48$ from both sides of the equation.

$3 s^{2} + 2 s + 1 - 48 = 0$

Step 4.1.2

Subtract $48$ from $1$ .

$3 s^{2} + 2 s - 47 = 0$

Step 4.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.3

Substitute the values $a = 3$ , $b = 2$ , and $c = - 47$ into the quadratic formula and solve for $s$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (3 \cdot - 47)}}{2 \cdot 3}$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Raise $2$ to the power of $2$ .

$s = \frac{- 2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 47}}{2 \cdot 3}$

Step 4.4.1.2

Multiply $- 4 \cdot 3 \cdot - 47$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $3$ .

$s = \frac{- 2 \pm \sqrt{4 - 12 \cdot - 47}}{2 \cdot 3}$

Step 4.4.1.2.2

Multiply $- 12$ by $- 47$ .

$s = \frac{- 2 \pm \sqrt{4 + 564}}{2 \cdot 3}$

Step 4.4.1.3

Add $4$ and $564$ .

$s = \frac{- 2 \pm \sqrt{568}}{2 \cdot 3}$

Step 4.4.1.4

Rewrite $568$ as $2^{2} \cdot 142$ .

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Step 4.4.1.4.1

Factor $4$ out of $568$ .

$s = \frac{- 2 \pm \sqrt{4 (142)}}{2 \cdot 3}$

Step 4.4.1.4.2

Rewrite $4$ as $2^{2}$ .

$s = \frac{- 2 \pm \sqrt{2^{2} \cdot 142}}{2 \cdot 3}$

Step 4.4.1.5

Pull terms out from under the radical.

$s = \frac{- 2 \pm 2 \sqrt{142}}{2 \cdot 3}$

Step 4.4.2

Multiply $2$ by $3$ .

$s = \frac{- 2 \pm 2 \sqrt{142}}{6}$

Step 4.4.3

Simplify $\frac{- 2 \pm 2 \sqrt{142}}{6}$ .

$s = \frac{- 1 \pm \sqrt{142}}{3}$

Step 4.5

The final answer is the combination of both solutions.

$s = - \frac{1 - \sqrt{142}}{3}, - \frac{1 + \sqrt{142}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$s = - \frac{1 - \sqrt{142}}{3}, - \frac{1 + \sqrt{142}}{3}$

Decimal Form:

$s = 3.63879176 \dots, - 4.30545842 \dots$